Probability: Concepts and Fundamentals
Definitions:
An event is any collection of results or outcomes from a set of procedures or trials.
A simple event is an event that has been broken down into its most basic parts.
Sample space – the listing of all possible outcomes of a procedure or trial. For example the
sample space for the genders of two children in two births would be {boy, boy; boy, girl; girl,
boy; girl, girl}.
Compliment – The compliment of any event is all other possible outcomes that exclude that
event. Notation: For Event 𝐴 its compliment is 𝐴̅.
Disjoint – Two events are disjoint if the possibility of one precludes the possibility of the other.
If one occurs, the other cannot possibly occur.
Mathematical Definitions of Probability:
Relative Frequency Approach - 𝑃(𝐴) =
Classical Approach - 𝑃(𝐴) =
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒 𝐸𝑣𝑒𝑛𝑡 𝐴 𝑜𝑐𝑐𝑢𝑟𝑒𝑑
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑣𝑒𝑛𝑡𝑠
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 𝐸𝑣𝑒𝑛𝑡 𝐴 𝑐𝑎𝑛 𝑜𝑐𝑐𝑢𝑟
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑖𝑚𝑝𝑙𝑒 𝑒𝑣𝑒𝑛𝑡𝑠
Law of Large Numbers:
As a procedure is repeated over and over, the observed probability will approach the actual
probability. For example, getting 10 tails in a 10 coin flips is an anomaly, but as we flip the coin
more and more, the probability of our procedure will get closer and closer to 50% heads and 50%
tails.
Unlikely vs. Unusual:
An event is unlikely if its chance of occurring is 5% or less [𝑃(𝑥) ≤ .05] whereas an outcome is
considered to be unusual if its results are extreme, either far above or below the expected value.
Independent vs. Dependent Events:
Two events are said to be independent of each other if the possibility of the first does not affect
the possibility of the second. Conversely, two events are dependent if the probability of the first
occurring does affect the probability of the second event occurring.
 Sampling with replacement means that events are independent.
 Conversely sampling without replacement means that events are dependent.
Middlesex Community College | Prepared by: Stephen McDonald